A331997 Semiprimes m = p*q where m, p and q are in A033638 (locations of right angle turns in Ulam square spiral).

10, 21, 26, 65, 91, 111, 785, 842, 1333, 4097, 21171, 28562, 50851, 100807, 194923, 970226, 1000001, 37021141, 65618101, 81144065, 151782401, 151819363, 174134417, 577921601, 688773781, 796622401, 796678851, 1276025563, 2090501285, 2176782337, 3662246773, 6059299123, 6879790081

Offset: 1

Author

Michel Lagneau, Feb 04 2020

Keywords

• nonn

Comments

The sequence is probably infinite.

A geometric property of the sequence: consider the first diagonal with numbers of the form f(k) = k^2 + k + 1 in the Ulam spiral. The semiprimes and their prime factors belonging to the diagonal are given by the subsequence: 21, 91, 1333, 50851, 194923, 37021141, 65618101, 151819363, 688773781, 796622401, 1276025563, 3662246773, 6059299123, 6879790081, ... (see the illustration). This subsequence is the result of the following property: f(k)*f(k+1) = f((k+1)^2).

Examples:

21 = 3*7 = f(1)*f(2) = f(4);

91 = 7*13 = f(2)*f(3) = f(9);

1333 = 31*43 = f(5)*f(6) = f(36);

................................

This subsequence is probably infinite.

Examples

			111 is in the sequence because 111 = 3*37, and the numbers 3, 37 and 111 are in A033638.
		

Links

• Michel Lagneau, Ulam Spiral

Crossrefs

Cf. A001358, A033638, A172979.

Programs

Maple

with(numtheory):nn:=10^5:T1:=1:
lst:={1}:lst1:={}:
  for n from 2 to nn do:
   T2:= T1 + floor(n/2):lst:=lst union {T2}:T1:=T2:
  od:
      for j from 2 to nn do:
        x:=lst[j]:d:=factorset(x):n0:=nops(d):
          if n0=2 and bigomega(x)=2
           and {d[1],d[2]} intersect lst = {d[1],d[2]}
           then
           lst1:=lst1 union {lst[j]}
           else
           fi:
          od:
          sort(lst1);

PARI

lista(nn) = {my(vn = vector(nn, k, k^2\4 + 1)); for (i=1, #vn, if (bigomega(vn[i]) == 2, my(f=factor(vn[i])); my(p=f[1,1], q = f[2,1]); if (vecsearch(vn, p) && vecsearch(vn, q), print1(vn[i], ", "));););} \\ Michel Marcus, Feb 04 2020

Extensions

More terms from Michel Marcus, Jan 13 2025